Born in Sydney, Briscoe began karting in 1993 and won Australian, North American, and Italian karting championships before moving to European single-seater racing in 2000. He won the Italian Formula Renault Championship in 2001 and the Formula Three Euroseries in 2003, then joined Toyota as an F1 test driver. Rather than moving to Formula One, he transferred to America, joining Target Chip Ganassi Racing in the IndyCar Series in 2005, where he was involved in a violent crash at Chicagoland Speedway that broke both clavicles and required eight weeks of recovery.
After a period with Dreyer and Reinbold Racing, Briscoe signed with Penske Racing as a full-time IndyCar driver for 2008. He won his first race at Milwaukee Mile that year and went on to collect eight IndyCar wins in total. In 2009 he led the IndyCar championship for most of the season, trading the points lead with Dario Franchitti and Scott Dixon, before a late crash at Motegi cost him the title; he finished the season third. Briscoe also became the first Australian to win the home race at Surfers Paradise in 2008. He took IndyCar pole for the 2012 Indianapolis 500, the first Australian to do so, and scored his final IndyCar podium that year at Sonoma. He made his final IndyCar appearances as a substitute driver in 2015.
Briscoe built a parallel sports car career alongside his IndyCar work. He drove Penske's Porsche RS Spyder LMP2 programme in the American Le Mans Series in 2007, taking three wins, and won the 2008 Petit Le Mans LMP2 class with Helio Castroneves. He returned to ALMS in 2013 with Level 5 Motorsports, winning the 12 Hours of Sebring in the LMP2 class. With Corvette Racing in 2015 he won the GTLM class at both the 24 Hours of Daytona and the 12 Hours of Sebring alongside Antonio Garcia and Jan Magnussen.
In 2016 Ford returned to endurance sports car racing with Chip Ganassi Racing, deploying the new Ford GT in both the FIA World Endurance Championship and the IMSA WeatherTech SportsCar Championship. Briscoe was selected for the IMSA team, where he partnered Richard Westbrook for full seasons, with Scott Dixon joining as an endurance driver at selected rounds including Le Mans.
Briscoe and Westbrook finished second in the GTLM Drivers' Championship in both 2016 and 2018, with their best Le Mans result being third in the GTE-Pro class in 2016. In 2018, the trio of Briscoe, Westbrook, and Dixon won the GTLM category at the 24 Hours of Daytona after 783 laps. The Ford factory programme concluded at the end of the 2019 season.
For the 2020 IMSA WeatherTech Championship, Briscoe moved to the Prototype class with Wayne Taylor Racing, partnering Renger van der Zande as the full-season crew. They were joined by Scott Dixon and Kamui Kobayashi for the 24 Hours of Daytona and won the race in record-breaking fashion, completing 833 laps. In October 2020, Briscoe, van der Zande, and Dixon also won the Petit Le Mans at Road Atlanta. Briscoe became a naturalized American citizen in April 2018.
Briscoe attended Trinity Grammar School in Sydney. In addition to English, he speaks Italian and French. He has been married to ESPN anchor Nicole Briscoe since 2009, and the couple have two children.